One or more embodiments of the invention relate in general to optical devices, such as optical modulators and related devices. In particular, one or more embodiments concern a (quasi) one-dimensional photonic crystal cavity, which can be configured as an all-optical modulator, or an all-optical transistor, as well as related devices and apparatuses. One or more embodiments further concern methods for modulating an optical signal, e.g., a data signal, using such a photonic crystal cavity.
Optical modulators are used for high-speed optical data and telecommunication systems to encode data into a stream of light pulses. In current technology, electrical signals are typically converted to optical signals through various types of electro-optical modulators.
Furthermore, photonic crystals, which are natural or artificial structures with periodic modulation of the refractive index, are known in the art. Depending on the geometry of their structure, photonic crystals can be categorized as one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) structures. In one-dimensional photonic crystals, the periodic modulation of the permittivity occurs in one direction only. Known examples of photonic crystals are Bragg gratings, which are commonly used as distributed reflectors in vertical cavity surface emitting lasers.
Furthermore, quasi one-dimensional photonic crystals are known, e.g., comprising a freestanding cavity that comprises holes and a defect at the center (see e.g., P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Lončar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009)). The central defect may be surrounded by tapered subsets of holes.
More generally, many optical devices are known, e.g., Fabry-Perot-like cavities and microcavities, which comprise an optical cavity. The geometry (including thickness or width) of the cavity determines the “cavity modes”, i.e., particular electromagnetic field patterns formed by light confined in the cavity. An ideal cavity would confine light indefinitely (that is, without loss). The deviations from this ideal paradigm are either intentional (e.g., outcoupling) or due to design or fabrication related limitations or imperfections (e.g., scattering). They are captured by the quality factor Q, which is proportional to the confinement time in units of the optical period. Another important descriptive parameter is the effective mode volume (V), which relates to the spatial extent of the optical mode present in the cavity. In general, the realization of practical devices requires maximizing the ratio Q/V, i.e., high values for Q and low values for V are important to increase light-matter interactions in processes such as spontaneous emission, nonlinear optical processes and strong coupling.
More specifically, the quality factor or Q factor is a dimensionless parameter that is inversely related to the degree of damping or dissipation of an oscillator or resonator. The value of Q is usually defined as 2π times the total energy stored in the structure, divided by the energy lost in a single oscillation cycle. In optics, and more generally for high values of Q, the following definition can be adopted:
                              Q          =                                                    f                r                                            Δ                ⁢                                                                  ⁢                f                                      =                                          ω                r                            Δω                                      ,                            Equation        ⁢                                  ⁢                  (          1          )                    where fr is the resonant frequency, Δf is the bandwidth, ωr=2πfr is the angular resonant frequency, and Δω is the angular bandwidth. A cavity providing a larger Q confines the photons for a longer time.
The definition of the effective mode volume is, in the literature, usually inspired from Purcell effect calculations, giving rise to:
                                          V            eff                    =                                    ∫                                                ɛ                  ⁡                                      (                    r                    )                                                  ⁢                                                                                                E                      ⁡                                              (                        r                        )                                                                                                  2                                ⁢                                  dr                  3                                                                                    ɛ                ⁡                                  (                                      r                    max                                    )                                            ⁢                              max                ⁡                                  (                                                                                                          E                        ⁡                                                  (                          r                          )                                                                                                            2                                    )                                                                    ,                            Equation        ⁢                                  ⁢                  (          2          )                    where:
∈(r) is the dielectric constant as a function of the spatial coordinate r,
rmax is the location of the maximum squared electric field, and
E(r)| is the electric field strength at the spatial coordinate r.
Eq. 2 is related to the spatial intensity enhancement factor at the point of maximum electric field intensity.
When a dimensionless effective mode volume is used (as assumed in the following description), the above formula need be multiplied by a factor (n(rmax)/λvac)3, where n(rmax) is the index of refraction at the location of the maximum electric field and λvac is the vacuum wavelength of the light.